Math 8340 Functional Analysis 2
Semester 2, 2016-2017
Course Lecturer
Dr. Judith Packer, Dept. of Mathematics
Tel: (303) 492-6979
Office: Math 227
Email: packer@colorado.edu
URL: http://spot.colorado.edu/~packer
Course Syllabus: For course syllabus, click here!
Course Information:
This course is meant to familiarize the student with sufficent knowledge of Hilbert space theory and
operator theory to lead to a proof of the spectral theorem for a normal operator
on a Hilbert space and some applications.
Topics to be covered include:
Definition of
Hilbert spaces and bounded operators on a Hilbert space, the adjoint of
an operator, self-adjoint, normal, and unitary operators, orthogonal
projections;
Banach algebras, spectrum and resolvent of an element in a
unital Banach algebra, the Gelfand representation theory for unital
commutative Banach algebras;
C*-algebras, the Gelfand-Naimark
Theorem for unital commutative C*-algebras,
the spectrum of a bounded operator on a Hilbert space
and the continuous functional calculus;
Projection-valued measures, definitions and basic properties,
spectral measures for a normal operator and the spectral theorem for a normal operator, the
Borel functional calculus;
Study of applications of the spectral theorem, which will be chosen from the following topics: examples involving multiplication operators and
the Fourier transform, types of spectrum: point, discrete, continuous,
and residual; applications of the spectral theorem: positive operators,
square roots of positive operators, partial isometries and the polar
decomposition for a bounded operator; compact operators: definitions and
basic properties, examples, including integral operators,
spectral properties of compact operators, the Fredholm alternative,
integral equations,
spectral theorem for compact normal operators, integral operators and Mercer's theorem,
the trace of certain integral operators.
Prerequisite:
Math 8330, Functional Analysis 1, or instructor permission.
Course Text:
We will use the text "A Course in Functional Analysis", by John Conway,
2nd Edition, Springer, 1997.
Assessment:
- Homework Assignments (every three weeks or so): 60 %
- Final project on topic related to course material, due beginning of May: 40 % .
- Please note the final exam period - Tuesday, May 9th, 2017, 1:30pm – 4:00pm
- This time will be used for student presentations.
Lecture Hours and Venue:
MWF, 1:00 p.m.- 1:50 p.m., ECCR 110.
Consultation Hours:
2 p.m. - 3 p.m., MWF Math 227, and by appointment.
Homework Assignments:
Assignments will be given every two weeks or so. Show your work.
- Assignment 1 , Due Friday, February 3, 2017.
- Assignment 2 , due due Monday, February 27, 2017.
- Assignment 3 , due Friday, March 17, 2017.
- Some solutions to Assignment 3.
- Assignment 4, due Monday, April 24, 2017.
Some Important Names associated with Functional Analysis :
S. Banach
J. Fourier
D. Hilbert
M. Frechet
I. Gelfand
H. Hahn
L. Schwartz
S. Sobolev
M. Stone
J. von Neumann
K. Weierstrass
Back to the home page of Judith A. Packer
Last modified January 12, 2017.